Theory of Equations
Theory of Equations tests relationships between coefficients and roots, formation and transformation of equations, and the nature of roots. Questions often ask you to solve a cubic, form an equation with prescribed roots, or use a conjugate-pair rule.
Vieta and root structure
For ax³+bx²+cx+d=0 with roots α,β,γ: α+β+γ=−b/a, αβ+βγ+γα=c/a, αβγ=−d/a. For a quadratic ax²+bx+c=0, sum=−b/a and product=c/a. If the coefficients are rational, a surd root p+√q (q not a square) forces p−√q to be a root; a complex root forces its conjugate. Descartes' rule of signs bounds the number of positive and negative real roots. Reciprocal equations have coefficients symmetric or anti-symmetric and are solved by dividing by x² and substituting u=x+1/x or u=x−1/x.
Worked example
Solve x³−6x²+11x−6=0. Testing x=1 gives 1−6+11−6=0, so x−1 is a factor. Division gives x²−5x+6=(x−2)(x−3). The roots are 1,2,3. Check Vieta: sum=6=−(−6)/1; pair sum=2+3+6=11; product=6=−(−6)/1.
Traps
The product of roots is −d/a, not d/a. Do not invoke conjugate pairs when coefficients are not rational. When transforming x→1/x, the constant term must be nonzero; otherwise a root at 0 is lost. Testing a root is not enough: complete the factorisation and check the remaining quadratic.